Thermodynamic equivalence of two-dimensional imperfect attractive Fermi and repulsive Bose gases

We consider two-dimensional imperfect attractive Fermi and repulsive Bose gases consisting of spinless point particles whose total interparticle interaction energy is represented by $a{N}^2\mathrm{/}2V$ with $a=-a_F\le 0$ for fermions, and $a=a_B\ge 0$ for bosons. We show that in spite of the attraction the thermodynamics of $d=2$ imperfect Fermi gas remains well defined for $0\le a_F\le a_0=h^2\mathrm{/}2\pi m$, and is exactly the same as the one of the repulsive imperfect Bose gas with $a_B=a_0-a_F$. In particular, for $a_F=a_0$ one observes the thermodynamic equivalence of the attractive imperfect Fermi gas and the ideal Bose gas.